11.1 Local Thermodynamic Equilibrium. 1. the electron and ion velocity distributions are Maxwellian,

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1 Section 11 LTE Basic treatments of stellar atmospheres adopt, as a starting point, the assumptions of Local Thermodynamic Equilibrium (LTE), and hydrostatic equilibrium. The former deals with the microscopic properties of the atoms, and we will discuss it here; the latter addresses the large-scale conditions (and applies throughout a normal star), and is discussed in Section Local Thermodynamic Equilibrium Fairly obviously, in Local Thermodynamic Equilibrium (LTE) it is assumed that all thermodynamic properties in a small volume have the thermodynamic equilibrium values at the local values of temperature and pressure. Specically, this applies to quantities such as the occupation numbers of atoms, the opacity, emissivity, etc. The LTE assumption is equivalent to stating that 1. the electron and ion velocity distributions are Maxwellian, dn(v) dv ( ) m 3/ { mv = n π k k for number density n of particles of mass m at kinetic temperature T k ; }. the photon source function is given by the Planck function at the local temperature (i.e., S ν = B ν, and j ν = k ν B ν ). 3. the excitation equilibrium is given by the Boltzmann equation n j = g { } j (Ej E i ) n i g i (11.1) 103

2 4. the ionization equilibrium is given by the Saha equation n e n,1 = g { },1 χ1,i (πme ) 3/ n 1,i g 1,i h 3 (11.) where 1, i,, 1 denote levels i, 1 in ionization stages 1,. One might augment this list with the perfect-gas equation of state, P = n, but since this applies under many circumstances where LTE doesn't hold, it's not usually mentioned in this context. If a process is purely collisional, conditions are, naturally, determined on a purely local basis locally, and LTE applies. We have already encountered one such situation where LTE is a good approximation: free-free emission results from a purely collisional process, justifying our adoption of S ν = B ν (Section 6). If radiation plays a role, then provided the photon and particle mean free paths are short compared to the length scales over which conditions change signicantly (i.e., if the opacity is high), then we can again ect LTE to be a reasonable assumption; this is a good approximation in stellar interiors. In stellar atmospheres the LTE approximation may be a poor one, as photon mean free paths are typically larger than those of particles. Thus one region can be aected by the radiation eld in another part of the atmosphere (e.g., a deeper, hotter region). As a rule of thumb, therefore, LTE is a poor approximation if the radiation eld is important in establishing the ionization and excitation equilibria (as in hot stars, for example). It's more likely to be acceptable when particle densities are high and the radiation eld is relatively weak; for stars, this means higher gravities (i.e., main-sequence stars rather than supergiants) and cooler eective temperatures. When LTE breaks down, we have a `non-lte' (nlte) situation, and level populations must be calculated assuming statistical equilibrium (section.3.). 11. The Saha Equation The Boltzmann Equation gives the relative populations of two bound levels i and j, in some initial (or `parent') ionization stage `1': n 1,j = g { } 1,j (E1,j E 1,i ) n 1,i g 1,i (11.1) where E 1,i & E 1,j are the level energies (measured from the ground state, E 1,1 = 0), and g 1,i & g 1,j are their statistical weights (J + 1, where J is the total angular-momentum quantum number). 104

3 To generalize the Boltzmann eqtn. to deal with collisional ionization to the next higher (or `daughter') ionization stage `', we identify the upper level j with a continuum state; n 1,j, the number of parent ions in excitation state j, then equates with n,1 (v), the number of ionized atoms where the detached electron has velocity v. (Note that ionization stages `1' and `' always represent any two consecutive stages for example, H 0 and H +, or C + and C 3+.) The total statistical weight of the ionized system is given by the combined statistical weights 1 of the newly created ion and the electron, i.e., g g e(v); while the relevant energy is the sum of the ionization energy and the kinetic energy of the free electron. Thus we have ( ) n,1(v) = gge(v) (χ1,i + 1 mev ) (11.3) n 1,i g 1,i where χ 1,i = E E 1,i is the ionization potential for level i in the parent species. An aside: The statistical weight of a free electron. The statistical weight of a free electron is just the probability of nding it in a specic cell of `phase space'. Since the state of a free particle is specied by three spatial coördinates x, y, z and three momentum coördinates p(x), p(y), p(z), the number of quantum states (for which the statistical weights are each 1) in an element of phase space, is given by dx dy dz dp(x) dp(y) dp(z) = dn g e(v) = dn h 3 = dx dy dz dp(x) dp(y) dp(z) h3 where h is Planck's constant and the factor arises because the electron has two possible spin states. The statistical weight per unit volume is thus dp(x) dp(y) dp(z) h3 for a single electron. However, there may be other free electrons, from other ions, which occupy some of the available states in the element of phase space dn. If the number density of electrons is n e then the eective volume available to a collisionally ejected electron is reduced by a factor 1/n e. Thus the statistical weight available to a single free electron is g e(v) = dp(x) dp(y) dp(z) n eh 3. Furthermore, if the velocity eld is isotropic, the `momentum volume' can be replaced simply by its counterpart in spherical coördinates, dp(x) dp(y) dp(z) = 4πp dp Using these results we can write eqtn. (11.3) as ( n en,1(v) = gi `χ1,i + mev ) 1 4πp dp n 1,i h 3 g 1,i 1 The statistical weight is a form of probability, and the probability of `A and B', P (A+B), is the product P (A)P (B). 105

4 but the momentum p = m ev; i.e., 1 mev = p /(m e), whence n en,1(v) n 1,i = g h 3 g 1,i n χ1,i o j ff p 4πp dp. m e Since we're interested in the ionization balance (not the velocity distribution of the ionized electrons), we integrate over velocity to obtain the total number of daughter ions: n en,1 n 1,i = g h 3 g 1,i n χ1,i o 4π Z We can then use result of a standard integral, Z 0 to obtain x ` a x dx = π/ `4a 3 0 j ff p p dp. m e n e n,1 n 1,i = g g 1,i { χ1,i } (πme ) 3/ h 3 (11.) This is one common form of the Saha Equation (often ressed in terms of the ground state of the parent ion, n 1,1 ) Partition functions The version of the Saha equation given in eqtn. (11.) relates populations in single states of excitation for each ion. Generally, we are more interested in the ratios of number densities of dierent ions summed over all states of excitation i.e., the overall ionization balance. We determine this by dening the partition function as U = n g n ( E n / ) (an easily evaluated function of T ), whence n e n = U (πm e ) 3/ { } χ1 n 1 U 1 h 3 (11.4) where we use χ 1, the ground-state ionization potential of the parent atom, as it is to this that the partition function is referred (i.e., E 1,1 0). Since the electron pressure is P e = n e we can also ress the Saha equation in the form n = U (πm e ) 3/ ( ) 5/ { } χ1 n 1 U 1 h 3 P e 106 (11.5)

5 An illustration: hydrogen The Balmer lines of hydrogen, widely observed as absorption lines in stellar spectra, arise through photoexcitation from the n = level of neutral hydrogen. To populate the n = level, we might suppose that we need temperatures such that E 1, = 10.eV; i.e., T 10 5 K. However, the Hα line strength peaks in A0 stars, which are much cooler than this (T 10 4 K). Why? Because we need to consider ionization as well as excitation. We therefore need to combine the Saha and Boltzmann equations to obtain the density of atoms in a given state of excitation, for a given state of ionization. We ress the Boltzmann equation, eqtn. (11.1), in terms of the partition function U: n 1, = g ( ) 1, E1, n 1 U 1 where n 1 is the number density of H 0 atoms in all excitation states and E 1, is the excitation energy of the n = level (10. ev); that is, n 1, = g ( ) 1, E1, n 1. U 1 However, the total number of hydrogen nuclei is n(h) = n 1 + n = n 1 (1 + n /n 1 ); that is, n 1 = n(h)(1 + n /n 1 ) 1. Using this, and n /n 1 from eqtn. (11.4), we nd n 1, = g 1, U 1 = g 1, U 1 ( E1, ( E1, ) ( [ U (πm e ) 3/ { } ]) 1 χ1 1 + n 1 U 1 h 3 n(h) ) ( 1 + [ U (πm e ) 3/ ( ) 5/ { } ]) 1 χ1 n 1 U 1 h 3 n(h) P e We can now see why the Balmer lines peak around 10 4 K: while higher temperatures give larger populations n 1, /n(h 0 ), they give smaller populations n(h 0 )/n(h). The overall result is that n 1, /n(h) peaks around 10kK. The Saha equation also gives an lanation of why supergiant stars are cooler than main-sequence stars of the same spectral type. Spectral characteristics are dened by ratios of lines strengths; e.g., O-star subtypes are dened by the ratio (He ii λ454)/(he i λ4471), which in turn traces the ratio He + /He 0. Of course, higher temperatures increase the latter ratio. However, a supergiant star has a lower surface gravity (and atmospheric pressure) than a main-sequence star. From eqtn. (11.5) we see that a lower pressure at the same temperature gives rise to a larger ratio n /n 1, so for two stars of the same temperature, the supergiant has an earlier spectral type (or, equivalently, at the same spectral type the supergiant is cooler). 107

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7 Section 1 Stellar Timescales 1.1 Dynamical timescale `Hydrostatic equilibrium' approach If we look at the Sun in detail, we see that there is vigorous convection in the envelope. With gas moving around, is the assumption of hydrostatic equilibrium justied? To address this question, we need to know how quickly displacements are restored; if this happens quickly (compared to the displacement timescales), then hydrostatic equilibrium remains a reasonable approximation even under dynamical conditions. We have written an appropriate equation of motion, ρ a = ρ g + dp dr (10.) where g is the acceleration due to gravity and a = d r dt is the nett acceleration. As the limiting case we can `take away' gas-pressure support (i.e., set dp /dr = 0), so our equation of motion for collapse under gravity is just d r dt = Gm(r) r. Integrating (and taking r from the surface inwards), r = Gm(r) r t = 1 gt (for initial velocity v 0 = 0). (1.1) 109

8 Identifying the time t in eqtn. (1.1) with a dynamical timescale, we have r t dyn = 3 Gm(r). (1.) Departures from hydrostatic equilibrium are restored on this timescale (by gravity in the case of ansion, or pressure in the case of contraction). In the case of the Sun, R 3 t dyn = 37 min. GM If you removed gas-pressure support from the Sun, this is how long it would take a particle at the surface to free-fall to the centre. Since the geological record shows that the Sun hasn't changed substantially for at least 10 9 yr, it is clear that any departures from hydrostatic equilibrium must be extremely small on a global scale. We might ect departures from spherical symmetry to be restored on a dynamical timescale (in the absence of signicant centrifugal forces), and by indeed comparing eqtns. (10.6) and (1.) we see that spherical symmetry is appropriate if ω t dyn 1.1. `Virial' approach The `hydrostatic equilibrium' approach establishes a collapse timescale for a particle to fall from the surface to the centre. As an alternative, we can consider a timescale for gas pressure to ll a void a pressure-support timescale. Noting that a pressure wave propagates at the sound speed, this dynamical timescale can be equated to a sound-crossing time for transmitting a signal from the centre to the surface. The sound speed is given by ( ) c S = γ µm(h) (1.3) (where γ = C p /C v, the ratio of specic heats at constant pressure and constant volume). From the virial theorem, U + Ω = 0 (10.11) with U = V 3 n(r) dv = V 3 ρ(r) dv (10.8) µm(h) 110

9 and so that M Gm(r) Ω = dm = 0 r V Gm(r) ρ(r) dv (10.10) r 3 µm(h) = Gm(r) ; r (1.4) that is, from eqtn. (1.3), 3 γ c S = Gm(r) r (1.5) For a monatomic gas we have γ = 5 /3, giving, from eqtns. (1.3) and (1.4), c S = 5 9 Gm(r) r so that the (centre-to-surface) sound crossing time is t = r 9/5r = 3 c S Gm(r) (1.6) (which is within 10% of eqtn. (1.)). 1. Kelvin-Helmholtz and Thermal Timescales Before nuclear fusion was understood, the conversion of potential to radiant energy, through gravitational contraction, was considered as a possible source of the Sun's luminosity. 1 The time over which the Sun's luminosity can be powered by this mechanism is the Kelvin-Helmholtz timescale. The available gravitational potential energy is Ω = M 0 Gm(r) r dm (10.10) but m(r) = 4 3 πr3 ρ so dm = 4πr ρ dr 1 Recall from Section 10.4 that half the gravitational potential energy lost in contraction is radiated away, with the remainder going into heating the star. 111

10 and Ω = R 0 G 16π 3 r4 ρ (r) dr π Gρ R 5 (1.7) (assuming ρ(r) = ρ(r)). The Kelvin-Helmholtz timescale for the Sun is therefore t KH = Ω( ) L. (1.8) which for ρ = kg m 3, Ω = J is t KH 10 7 yr. The Kelvin-Helmholtz timescale is often identied with the thermal timescale, but the latter is more properly dened as t th = U( ) L, (1.9) which (from the virial theorem) is 1 /t KH. In practice, the factor dierence is of little importance for these order-of-magnitude timescales. 1.3 Nuclear timescale We now know that the source of the Sun's energy is nuclear fusion, and we can calculate a corresponding nuclear timescale, t N = fmc L (1.10) where f is just the fraction of the rest mass available to the relevant nuclear process. In the case of hydrogen burning this fractional `mass defect' is 0.007, so we might ect t N = 0.007M c ( yr for the Sun). L However, in practice, only the core of the Sun about 10% of its mass takes part in hydrogen burning, so its nuclear timescale for hydrogen burning is yr. Other evolutionary stages have their respective (shorter) timescales. At the time that this estimate was made, the fossil record already indicated a much older Earth ( 10 9 yr). Kelvin noted this discrepancy, but instead of rejecting contraction as the source of the Sun's energy, he instead chose to reject the fossil record as an indicator of age. 11

11 1.4 Diusion timescale for radiative transport Deep inside stars the radiation eld is very close to black body. For a black-body distribution the average photon energy is E = U/n T [J photon 1 ]. (1.8) The core temperature of the Sun is T c 1 1 / 10 7 K (cf. eqtn ), whence E = 3.5 kev i.e., photon energies are in the X-ray regime. Light escaping the surface of the Sun (T eff 5770K) has a mean photon energy smaller, in the optical. The source of this degradation in the mean energy is the coupling between radiation and matter. Photons obviously don't ow directly out from the core, but rather they diuse through the star, travelling a distance of order the local mean free path, l, before being absorbed and re-emitted in some other direction (a `random walk'). The mean free path depends on the opacity of the gas: l = 1/k ν = 1/(κ ν ρ) (.4) where k ν is the volume opacity (units of area per unit volume, or m 1 ) and κ ν is the mass opacity (units of m kg 1 ). After n sc scatterings the radial distance travelled is, on average, n sc l (it's a statistical, random-walk process). Thus to travel a distance R we require ( ) R n sc =. (1.11) l Solar-structure models give an average mean free path l 1 mm (incidentally, justifying the LTE approximation in stellar interiors); with R m, n sc The total distance travelled by a (ctitious!) photon travelling from the centre to the surface is n sc l m ( 10 1 R!), and the time to diuse to the surface is (n sc l)/c yr. [More detailed calculations give yr; why? Naturally, there are regions within the Sun that have greater or lesser opacity than the average value, with the largest opacities in the central 0.4R and in the region immediately below the photosphere. Because of the `square root' nature of the diusion, a region with twice the opacity takes four times longer to pass through, while a region with half the opacity takes only times shorter; so any non-uniformity in the opacity inevitably leads to a longer total diusion time.] 113